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Mirrors > Home > MPE Home > Th. List > Mathboxes > crngocom | Structured version Visualization version GIF version |
Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.) |
Ref | Expression |
---|---|
crngocom.1 | ⊢ 𝐺 = (1st ‘𝑅) |
crngocom.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
crngocom.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
crngocom | ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngocom.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | crngocom.2 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | crngocom.3 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | iscrngo2 32966 | . . . 4 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥))) |
5 | 4 | simprbi 479 | . . 3 ⊢ (𝑅 ∈ CRingOps → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) |
6 | oveq1 6556 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦)) | |
7 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐻𝑥) = (𝑦𝐻𝐴)) | |
8 | 6, 7 | eqeq12d 2625 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝐴𝐻𝑦) = (𝑦𝐻𝐴))) |
9 | oveq2 6557 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵)) | |
10 | oveq1 6556 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐻𝐴) = (𝐵𝐻𝐴)) | |
11 | 9, 10 | eqeq12d 2625 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝑦𝐻𝐴) ↔ (𝐴𝐻𝐵) = (𝐵𝐻𝐴))) |
12 | 8, 11 | rspc2v 3293 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))) |
13 | 5, 12 | mpan9 485 | . 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
14 | 13 | 3impb 1252 | 1 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ran crn 5039 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 RingOpscrngo 32863 CRingOpsccring 32962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-1st 7059 df-2nd 7060 df-rngo 32864 df-com2 32959 df-crngo 32963 |
This theorem is referenced by: crngm23 32971 crngohomfo 32975 isidlc 32984 dmncan2 33046 |
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